114 research outputs found
Out-of-Time-Order Correlation at a Quantum Phase Transition
In this paper we numerically calculate the out-of-time-order correlation
functions in the one-dimensional Bose-Hubbard model. Our study is motivated by
the conjecture that a system with Lyapunov exponent saturating the upper bound
will have a holographic dual to a black hole at finite
temperature. We further conjecture that for a many-body quantum system with a
quantum phase transition, the Lyapunov exponent will have a peak in the quantum
critical region where there exists an emergent conformal symmetry and is absent
of well-defined quasi-particles. With the help of a relation between the
R\'enyi entropy and the out-of-time-order correlation function, we argue that
the out-of-time-order correlation function of the Bose-Hubbard model will also
exhibit an exponential behavior at the scrambling time. By fitting the
numerical results with an exponential function, we extract the Lyapunov
exponents in the one-dimensional Bose-Hubbard model across the quantum critical
regime at finite temperature. Our results on the Bose-Hubbard model support the
conjecture. We also compute the butterfly velocity and propose how the echo
type measurement of this correlator in the cold atom realizations of the
Bose-Hubbard model without inverting the Hamiltonian.Comment: 7 pages, 6 figures, published versio
Out-of-Time-Order Correlation for Many-Body Localization
In this paper we first compute the out-of-time-order correlators (OTOC) for
both a phenomenological model and a random-field XXZ model in the many-body
localized phase. We show that the OTOC decreases in power law in a many-body
localized system at the scrambling time. We also find that the OTOC can also be
used to distinguish a many-body localized phase from an Anderson localized
phase, while a normal correlator cannot. Furthermore, we prove an exact theorem
that relates the growth of the second R\'enyi entropy in the quench dynamics to
the decay of the OTOC in equilibrium. This theorem works for a generic quantum
system. We discuss various implications of this theorem.Comment: 6 pages, 3 figures, published versio
Generalized Real-space Chern Number Formula and Entanglement Hamiltonian
We generalize the real-space Chern number formula for gapped free fermion
Hamiltonians. Using this generalized formula, we prove the recent proposals for
extracting thermal and electric Hall conductance from the ground state via
entanglement Hamiltonian, in the special case of Gaussian states.Comment: 9 pages + 3 appendice
Periodically, Quasi-periodically, and Randomly Driven Conformal Field Theories (II): Furstenberg's Theorem and Exceptions to Heating Phases
In this sequel (to [Phys. Rev. Res. 3, 023044(2021)], arXiv:2006.10072), we
study randomly driven dimensional conformal field theories (CFTs), a
family of quantum many-body systems with soluble non-equilibrium quantum
dynamics. The sequence of driving Hamiltonians is drawn from an independent and
identically distributed random ensemble. At each driving step, the deformed
Hamiltonian only involves the energy-momentum density spatially modulated at a
single wavelength and therefore induces a M\"obius transformation on the
complex coordinates. The non-equilibrium dynamics is then determined by the
corresponding sequence of M\"obius transformations, from which the Lyapunov
exponent is defined. We use Furstenberg's theorem to classify the
dynamical phases and show that except for a few \emph{exceptional points} that
do not satisfy Furstenberg's criteria, the random drivings always lead to a
heating phase with the total energy growing exponentially in the number of
driving steps and the subsystem entanglement entropy growing linearly in
with a slope proportional to central charge and the Lyapunov exponent
. On the contrary, the subsystem entanglement entropy at an
exceptional point could grow as while the total energy remains to
grow exponentially. In addition, we show that the distributions of the operator
evolution and the energy density peaks are also useful characterizations to
distinguish the heating phase from the exceptional points: the heating phase
has both distributions to be continuous, while the exceptional points could
support finite convex combinations of Dirac measures depending on their
specific type. In the end, we compare the field theory results with the lattice
model calculations for both the entanglement and energy evolution and find
remarkably good agreement.Comment: 66 pages, 1 table, many figures; This is part II on randomly driven
CFT
Emergent Spatial Structure and Entanglement Localization in Floquet Conformal Field Theory
We study the energy and entanglement dynamics of D conformal field
theories (CFTs) under a Floquet drive with the sine-square deformed (SSD)
Hamiltonian. Previous work has shown this model supports both a non-heating and
a heating phase. Here we analytically establish several robust and
`super-universal' features of the heating phase which rely on conformal
invariance but not on the details of the CFT involved. First, we show the
energy density is concentrated in two peaks in real space, a chiral and
anti-chiral peak, which leads to an exponential growth in the total energy. The
peak locations are set by fixed points of the M\"obius transformation. Second,
all of the quantum entanglement is shared between these two peaks. In each
driving period, a number of Bell pairs are generated, with one member pumped to
the chiral peak, and the other member pumped to the anti-chiral peak. These
Bell pairs are localized and accumulate at these two peaks, and can serve as a
source of quantum entanglement. Third, in both the heating and non-heating
phases we find that the total energy is related to the half system entanglement
entropy by a simple relation with being the central charge. In addition, we show that the
non-heating phase, in which the energy and entanglement oscillate in time, is
unstable to small fluctuations of the driving frequency in contrast to the
heating phase. Finally, we point out an analogy to the periodically driven
harmonic oscillator which allows us to understand global features of the
phases, and introduce a quasiparticle picture to explain the spatial structure,
which can be generalized to setups beyond the SSD construction.Comment: 41 pages, 19 figure
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